A marbles of diameter 1.4 cm are dropped into a cylinder beaker containing some water and are fully submerged. The diameter of the beaker is 7cm. Find how many marbles have been dropped in it , if the water rises by 5.6 cm ?

Options

A. 50
B. 150
C. 500
D. 550

Answer

To solve the problem of how many marbles are dropped into the beaker, we will follow these steps:

### Step 1: Calculate the volume of one marble.
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Given the diameter of the marble is 1.4 cm, we can find the radius \( r \):
\[ r = \frac{\text{diameter}}{2} = \frac{1.4}{2} = 0.7 \text{ cm} \]
Now, substituting the radius into the volume formula:
\[ V = \frac{4}{3} \pi (0.7)^3 \]
Calculating \( (0.7)^3 \):
\[ (0.7)^3 = 0.343 \]
Now substituting this value back:
\[ V = \frac{4}{3} \pi (0.343) \approx \frac{4 \times 0.343 \times 3.14}{3} \approx \frac{4.305}{3} \approx 1.435 \text{ cm}^3 \]

### Step 2: Calculate the volume of water displaced in the beaker.
The volume of water displaced can be calculated using the formula for the volume of a cylinder:
\[ V = \pi r^2 h \]
The diameter of the beaker is 7 cm, so the radius \( r \) is:
\[ r = \frac{7}{2} = 3.5 \text{ cm} \]
The height \( h \) by which the water rises is given as 5.6 cm. Now substituting these values into the volume formula:
\[ V = \pi (3.5)^2 (5.6) \]
Calculating \( (3.5)^2 \):
\[ (3.5)^2 = 12.25 \]
Now substituting this value back:
\[ V = \pi (12.25)(5.6) \approx 68.6 \pi \text{ cm}^3 \]

### Step 3: Calculate the number of marbles.
To find the number of marbles \( n \), we divide the volume of water displaced by the volume of one marble:
\[ n = \frac{\text{Volume of water displaced}}{\text{Volume of one marble}} = \frac{68.6 \pi}{1.435} \]
Calculating this gives:
\[ n \approx \frac{68.6 \times 3.14}{1.435} \approx \frac{215.524}{1.435} \approx 150 \]

### Final Answer:
Thus, the number of marbles that have been dropped into the beaker is approximately **150 marbles**.

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